The Experts below are selected from a list of 60 Experts worldwide ranked by ideXlab platform
Christiane Kraus - One of the best experts on this subject based on the ideXlab platform.
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A degenerating Cahn–Hilliard system coupled with complete damage processes
Nonlinear Analysis: Real World Applications, 2015Co-Authors: Christian Heinemann, Christiane KrausAbstract:In this work, we analytically investigate a degenerating PDE system for phase separation and complete damage processes considered on a Nonsmooth time-dependent Domain with mixed boundary conditions. The evolution of the system is described by a degenerating Cahn–Hilliard equation for the concentration, a doubly nonlinear differential inclusion for the damage variable and a degenerating quasi-static balance equation for the displacement field. All these equations are highly nonlinearly coupled. Because of the doubly degenerating character of the system, the doubly nonlinear differential inclusion and the Nonsmooth Domain, the structure of the model is very complex from an analytical point of view. A novel approach is introduced for proving existence of weak solutions for such degenerating coupled system. To this end, we first establish a suitable notion of weak solutions, which consists of weak formulations of the diffusion and the momentum balance equation, a variational inequality for the damage process and a total energy inequality. To show existence of weak solutions, several new ideas come into play. Various results on shrinking sets and its corresponding local Sobolev spaces are used. It turns out that, for instance, on open sets which shrink in time a quite satisfying analysis in Sobolev spaces is possible. The presented analysis can handle highly Nonsmooth regions where complete damage takes place. To mention only one difficulty, infinitely many completely damaged regions which are not connected with the Dirichlet boundary may occur in arbitrary small time intervals.
Hussein Cheikh Ali - One of the best experts on this subject based on the ideXlab platform.
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Hardy–Sobolev inequalities with singularities on non smooth boundary: Hardy constant and extremals. Part I: Influence of local geometry
Nonlinear Analysis, 2019Co-Authors: Hussein Cheikh AliAbstract:Abstract Let Ω be a Domain of R n , n ≥ 3 . The classical Caffarelli–Kohn–Nirenberg inequality rewrites as the following inequality: for any s ∈ [ 0 , 2 ] and any γ ( n − 2 ) 2 4 , there exists a constant K ( Ω , γ , s ) > 0 such that ( H S ) ∫ Ω | u | 2 ⋆ ( s ) | x | s d x 2 2 ⋆ ( s ) ≤ K ( Ω , γ , s ) ∫ Ω | ∇ u | 2 − γ u 2 | x | 2 d x , for all u ∈ D 1 , 2 ( Ω ) (the completion of C c ∞ ( Ω ) for the relevant norm). When 0 ∈ Ω is an interior point, the range ( − ∞ , ( n − 2 ) 2 4 ) for γ cannot be improved: moreover, the optimal constant K ( Ω , γ , s ) is independent of Ω and there is no extremal for ( H S ) . But when 0 ∈ ∂ Ω , the situation turns out to be drastically different since the geometry of the Domain impacts : • the range of γ ’s for which ( H S ) holds. • the value of the optimal constant K ( Ω , γ , s ) ; • the existence of extremals for ( H S ) . When Ω is smooth, the problem was tackled by Ghoussoub–Robert (2017) where the role of the mean curvature was central. In the present paper, we consider Nonsmooth Domain with a singularity at 0 modeled on a cone. We show how the local geometry induced by the cone around the singularity influences the value of the Hardy constant on Ω . When γ is small, we introduce a new geometric object at the conical singularity that generalizes the ”mean curvature”: this allows to get extremals for ( H S ) . The case of larger values for γ will be dealt in the forthcoming paper (Cheikh-Ali, 2018). As an intermediate result, we prove the symmetry of some solutions to singular pdes that has an interest on its own.
Christian Heinemann - One of the best experts on this subject based on the ideXlab platform.
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A degenerating Cahn–Hilliard system coupled with complete damage processes
Nonlinear Analysis: Real World Applications, 2015Co-Authors: Christian Heinemann, Christiane KrausAbstract:In this work, we analytically investigate a degenerating PDE system for phase separation and complete damage processes considered on a Nonsmooth time-dependent Domain with mixed boundary conditions. The evolution of the system is described by a degenerating Cahn–Hilliard equation for the concentration, a doubly nonlinear differential inclusion for the damage variable and a degenerating quasi-static balance equation for the displacement field. All these equations are highly nonlinearly coupled. Because of the doubly degenerating character of the system, the doubly nonlinear differential inclusion and the Nonsmooth Domain, the structure of the model is very complex from an analytical point of view. A novel approach is introduced for proving existence of weak solutions for such degenerating coupled system. To this end, we first establish a suitable notion of weak solutions, which consists of weak formulations of the diffusion and the momentum balance equation, a variational inequality for the damage process and a total energy inequality. To show existence of weak solutions, several new ideas come into play. Various results on shrinking sets and its corresponding local Sobolev spaces are used. It turns out that, for instance, on open sets which shrink in time a quite satisfying analysis in Sobolev spaces is possible. The presented analysis can handle highly Nonsmooth regions where complete damage takes place. To mention only one difficulty, infinitely many completely damaged regions which are not connected with the Dirichlet boundary may occur in arbitrary small time intervals.
Hussein Cheikh Ali - One of the best experts on this subject based on the ideXlab platform.
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Hardy-Sobolev inequalities with singularities on non smooth boundary: Hardy constant and extremals. Part I: Influence of local geometry
Nonlinear Analysis: Theory Methods and Applications, 2019Co-Authors: Hussein Cheikh AliAbstract:Let $\Omega$ be a Domain of $\mathbb{R}^n$, $n\geq 3$. The classical Caffarelli-Kohn-Nirenberg inequality rewrites as the following inequality: for any $s\in [0,2]$ and any $\gamma0$ such that $$\left(\int_{\Omega}\frac{|u|^{\crit}}{|x|^s}\, dx\right)^{\frac{2}{\crit}}\leq K(\Omega,\gamma,s)\int_{\Omega}\left(|\nabla u|^2-\gamma\frac{u^2}{|x|^2}\right)\, dx,\eqno{(HS)}$$ for all $u\in D^{1,2}(\Omega)$ (the completion of $C^\infty_c(\Omega)$ for the relevant norm). When $0\in\Omega$ is an interior point, the range $(-\infty, \frac{(n-2)^2}{4})$ for $\gamma$ cannot be improved: moreover, the optimal contant $K(\Omega,\gamma,s)$ is independent of $\Omega$ and there is no extremal for $(HS)$. But when $0\in\partial\Omega$, the situation turns out to be drastically different since the geometry of the Domain impacts : \begin{itemize} \item the range of $\gamma$'s for which $(HS)$ holds; \item the value of the optimal constant $K(\Omega,\gamma,s)$; \item the existence of extremals for $(HS)$. \end{itemize} When $\Omega$ is smooth, the problem was tackled by Ghoussoub-Robert \cite{GR} where the role of the mean curvature was central. In the present paper, we consider Nonsmooth Domain with a singularity at $0$ modeled on a cone. We show how the local geometry induced by the cone around the singularity influences the value of the Hardy constant on $\Omega$. When $\gamma$ is small, we introduce a new geometric object at the conical singularity that generalizes the "mean curvature": this allows to get extremals for $(HS)$. The case of larger values for $\gamma$ will be dealt in the forthcoming paper \cite{HCA2}. As an intermediate result, we prove the symmetry of some solutions to singular pdes that has an interest on its own.
Michael Christ - One of the best experts on this subject based on the ideXlab platform.
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compactness in the neumann problem magnetic schrodinger operators and the aharonov bohm effect
Advances in Mathematics, 2005Co-Authors: Michael ChristAbstract:Abstract Compactness of the Neumann operator in the ∂ ¯ -Neumann problem is studied for weakly pseudoconvex bounded Hartogs Domains in two dimensions. A Nonsmooth Domain is constructed for which condition (P) fails to hold, yet the Kohn Laplacian still has compact resolvent. The main result, in contrast, is that for smoothly bounded Hartogs Domains, the well-known sufficient condition (P) is equivalent to compactness. The analyses of compactness and condition (P) boil down to the asymptotic behavior of the lowest eigenvalues of two related sequences of Schrodinger operators, one with a magnetic field and one without, parametrized by a Fourier variable resulting from the Hartogs symmetry. The Nonsmooth example is based on the Aharonov–Bohm phenomenon of quantum physics. For smooth Domains not satisfying (P), we prove that there always exists an exceptional sequence of Fourier variables for which the Aharonov–Bohm effect is weak and thence that compactness fails to hold. This sequence can be very sparse, so that the lack of compactness is due to a rather subtle effect.
